Decomposition of Multi-Exponential and Related Signals – Functional Filtering Approach

Decomposition of multi-exponential and related signals is generalized as an inverse filtering problem on a logarithmic time or frequency scale, and discrete-time filters operating with equally spaced data on a logarithmic scale (geometrically spaced on linear scale) are proposed for its implementation. Ideal prototypes, algorithms and types of filters are found for various timeand frequency-domain mono-components. It is disclosed that the ill-posedness in the decomposition originates as high sampling-rate dependent noise amplification coefficients arising from the large areas under the increasing frequency responses. A novel regularization method is developed based on the noise transformation regulation by filter bandwidth control, which is implemented by adaptation of the appropriate sampling rate. Algorithm design of decomposition filters is suggested joining together signal acquisition, regularization and discrete-time filter implementation. As an example, decomposition of a frequency-domain multi-component signal is considered by a designed filter. Key-Words: Decomposition, Multi-Component Signals, Distribution of Time Constants, Functional Filters, Logarithmic Sampling, Ill-posedness, Regularization